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# Mathematical Analysis of Projectile Motion

Consider a boy throwing a tennis ball in air through an angle; in a range that is fructose induced. Figure 1below shows the path taken by the ball through space. In our case scenario, the tennis ball is considered as a projectile that is two-dimensional. Such is due to the fact that it flies through air horizontally as well as vertically (KhanAcademy, 2018). However, gravity exerts its pressure on it. In this case, gravity is also considered as air resistance. By convention, gravitational force acts on objects moving through air. Mostly when dealing with projectile motion, air resistance is considered as negligible. In as much as such approximation is not a good idea, dealing with air resistance is tricky and thus ignoring it is appropriate. Also, negligence works well with object weights that exceed air resistance force. However, one should understand that air resistance on an object on motion increases with increase in speed (velocity). In this case, our discussion will consider instances where the velocity of an object is small and thus, air resistance negligible (KhanAcademy, 2018).

Back to our initial case, when the boy throws the ball, gravitational force pulls the ball to earth. Such means that gravity only influences the vertical component of the ball’s velocity (Vy). However, the horizontal movement of the ball (Vx) will remain unaffected by gravity. Vx remains constant as the ball moves in its path (KhanAcademy, 2018).

## Figure 1: Path taken by tennis ball

### Mathematical Handling of 2D Projectile

Dealing with 2D projectile motion mathematically involves separate analysis of vertical and horizontal motions. In this case, we shall use two sets of equations (KhanAcademy, 2018). One will describe the ball’s horizontal motion while the other describes the ball’s vertical motion. Therefore, a single 2D projectile problem that is difficult in nature will be turned into two 1D projectile problems that are simpler to work out. We are capable of working out the different 1D problems because changes in the ball’s vertical won’t affect its horizontal velocity. Also, if the ball would be thrown with a large horizontal velocity, its vertical acceleration would not be affected. In a different scenario, if a chalk is dropped and thrown horizontally, both will hit the ground within same time frames (KhanAcademy, 2018).

## 2D Projectile Problems

While calculating problems relating to 2D projectile motion, many people substitute horizontal components to vertical components of the two 1D equations and vice versa. However, it is important to note that independent analysis of the projectile will work if only the two directions are kept independent. Also, if the initial velocities of a projectile are diagonal in nature, breaking them into horizontal and vertical components is important. In such a case, there are those who find problems breaking down vector velocities into horizontal and vertical components (KhanAcademy, 2018).

## i) Horizontal movement

Since gravitational force does not work sideways, there is no acceleration in this direction. Because we are considering a negligible amount of air resistance, we assume that velocity is constant.

## ii) Vertical movement

For a 2D projectile, gravitational acceleration is experienced. Because our acceleration vertically will be constant, solving our vertical variable will involve the use of any of the four highlighted kinematic formulas in figure 2 below. However, one should be sure to only replace variables for vertical movement with those of the highlighted vertical equations. By knowing only three variables of a given vertical equation, solving the remaining variables that are unknown would be easy (KhanAcademy, 2018).

## Reference

KhanAcademy. (2018). what is 2D projectile motion?. Retrieved March 15, 2018, from https://www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/a/what-is-2d-projectile-motion